# Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

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## Abstract

**:**

## 1. Introduction and Governing Equation

## 2. Quintic B-spline Galerkin Method

## 3. The Initial Vectors

## 4. Numerical Experiments and Results

#### 4.1. Problem 4.1 (Single Solitary Wave Solution)

#### 4.2. Problem 4.2 (The Interaction of Two Solitary Waves)

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Interaction of two-solitons at different times with amplitude = 1, $\alpha =2$ and $\Delta t=0.0005.$

**Table 1.**Norms and conservation laws for Problem 4.1. with $\Delta t=0.002,\alpha =2,S=4\mathrm{and}\beta =1$.

$\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |
---|---|---|---|---|

$0.5$ | $1.9568\times {10}^{-4}$ | $2.7670\times {10}^{-4}$ | $1.99983872$ | $7.33354808$ |

$1.0$ | $1.9568\times {10}^{-4}$ | $2.7670\times {10}^{-4}$ | $1.99983872$ | $7.33354808$ |

$1.5$ | $1.9568\times {10}^{-4}$ | $2.7670\times {10}^{-4}$ | $1.99983872$ | $7.33354808$ |

$2.0$ | $1.9568\times {10}^{-4}$ | $2.7670\times {10}^{-4}$ | $1.99983872$ | $7.33354808$ |

$2.5$ | $1.9568\times {10}^{-4}$ | $2.7725\times {10}^{-4}$ | $1.99983871$ | $7.33354806$ |

$3.0$ | $6.4457\times {10}^{-4}$ | $3.0534\times {10}^{-4}$ | $1.99983850$. | $7.33354709$ |

$3.5$ | $4.7627\times {10}^{-3}$ | $9.9330\times {10}^{-4}$ | $1.99982692$ | $7.33349360$ |

**Table 2.**Norms and conservation laws for Problem 4.1 with $\Delta t=0.001,\alpha =2,S=4\mathrm{and}\beta =1$.

$\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |
---|---|---|---|---|

0.5 | $9.78400\times {10}^{-5}$ | $1.3835\times {10}^{-4}$ | $1.99991936$ | $7.33344071$ |

1.0 | $9.78400\times {10}^{-5}$ | $1.3835\times {10}^{-4}$ | $1.99991936$ | $7.33344071$ |

1.5 | $9.78400\times {10}^{-5}$ | $1.3835\times {10}^{-4}$ | $1.99991936$ | $7.33344071$ |

2.0 | $9.78400\times {10}^{-5}$ | $1.3837\times {10}^{-4}$ | $1.99991936$ | $7.33344071$ |

2.5 | $9.78400\times {10}^{-5}$ | $1.3945\times {10}^{-4}$ | $1.99991935$ | $7.33344069$ |

3.0 | $6.44590\times {10}^{-4}$ | $1.8914\times {10}^{-4}$ | $1.99991914$ | $7.33343971$ |

3.5 | $4.76289\times {10}^{-3}$ | $9.6293\times {10}^{-4}$ | $1.99990755$ | $7.33338623$ |

**Table 3.**Norms and conservation laws for Problem 4.1. with $\Delta t=0.005,\alpha =2,S=4\mathrm{and}\beta =1\mathrm{and}t=1$.

$\mathit{h}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |
---|---|---|---|---|

0.4000 | $3.15055\times {10}^{-3}$ | $4.36807\times {10}^{-3}$ | $1.99889027$ | $7.32694758$ |

0.3000 | $1.93900\times {10}^{-3}$ | $2.70993\times {10}^{-3}$ | $1.99828775$ | $7.32875439$ |

0.2667 | $1.62336\times {10}^{-3}$ | $2.28584\times {10}^{-3}$ | $1.99826101$ | $7.32850737$ |

0.2500 | $1.48026\times {10}^{-3}$ | $2.09864\times {10}^{-3}$ | $1.99827987$ | $7.32841934$ |

0.2000 | $1.11080\times {10}^{-3}$ | $1.60406\times {10}^{-3}$ | $1.99845175$ | $7.32849099$ |

0.1600 | $8.72850\times {10}^{-4}$ | $1.27057\times {10}^{-3}$ | $1.99869795$ | $7.32900596$ |

0.1333 | $7.38540\times {10}^{-4}$ | $1.07534\times {10}^{-3}$ | $1.99890569$ | $7.32957727$ |

0.1000 | $5.95490\times {10}^{-4}$ | $8.59430\times {10}^{-4}$ | $1.99920350$ | $7.33051338$ |

0.0800 | $5.19280\times {10}^{-4}$ | $7.42820\times {10}^{-4}$ | $1.99939758$ | $7.33117069$ |

0.0667 | $4.69970\times {10}^{-4}$ | $6.68230\times {10}^{-4}$ | $1.99953162$ | $7.33163940$ |

0.0500 | $4.04610\times {10}^{-4}$ | $5.72190\times {10}^{-4}$ | $1.99970290$ | $7.33225097$ |

0.0400 | $3.58160\times {10}^{-4}$ | $5.05630\times {10}^{-4}$ | $1.99980701$ | $7.33262780$ |

**Table 4.**Norms and order of convergence at $t=1$ with $h=\Delta t,\alpha =2,S=4\mathrm{and}\beta =1.$

$\mathit{h}$ | ${\mathit{L}}_{\mathit{\infty}}$ | Order | ${\mathit{L}}_{2}$ | Order |
---|---|---|---|---|

0.040 | $3.02045\times {10}^{-3}$ | - | $3.0881\times {10}^{-3}$ | - |

0.020 | $7.61510\times {10}^{-4}$ | $1.989$ | $7.7753\times {10}^{-4}$ | $1.9897$ |

0.010 | $1.90750\times {10}^{-4}$ | $1.997$ | $1.9473\times {10}^{-4}$ | $1.9974$ |

0.005 | $4.77100\times {10}^{-5}$ | $1.999$ | $4.8700\times {10}^{-5}$ | $1.9994$ |

**Table 5.**Comparison of present results for Problem 4.1 with different methods ($\alpha =2,S=4,\beta =1$, $t=1).$

$\mathit{h}$ | $\mathbf{\Delta}\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}(\mathbf{Galerkin}\mathbf{Quintic}\mathbf{B}\mathbf{-}\mathbf{spline}\mathbf{Method})$ | ${\mathit{L}}_{\mathit{\infty}}$ [17,18,19,21,22] |
---|---|---|---|

0.0300 | 0.0050 | $4.80\times {10}^{-4}$ | $3.00\times {10}^{-4}$ |

0.0500 | 0.0010 | $1.00\times {10}^{-4}$ | $5.77\times {10}^{-3}$ |

0.0500 | 0.0050 | $5.10\times {10}^{-4}$ | $3.00\times {10}^{-4}$ |

0.0600 | 0.0165 | $1.74\times {10}^{-3}$ | $1.50\times {10}^{-3}$ |

0.3125 | 0.0200 | $8.23\times {10}^{-3}$ | $2.00\times {10}^{-3}$ |

0.3125 | 0.0026 | $1.07\times {10}^{-3}$ | $5.13\times {10}^{-3}$ |

**Table 6.**Comparison of present results for Problem 4.1 with those of Taha et al. [21] $(\alpha =2,S=4,\beta =2$, $t=1).$

$\mathit{h}$ | $\mathbf{\Delta}\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}(\mathbf{Galerkin}\mathbf{Quintic}\mathbf{B}\mathbf{-}\mathbf{spline}\mathbf{Method})$ | ${\mathit{L}}_{\mathit{\infty}}$ [21] |
---|---|---|---|

$0.0300$ | $0.00022$ | $4.00\times {10}^{-5}$ | $7.59\times {10}^{-3}$ |

$0.1563$ | $0.00480$ | $1.62\times {10}^{-3}$ | $4.64\times {10}^{-3}$ |

$0.0700$ | $0.01200$ | $2.31\times {10}^{-3}$ | $9.37\times {10}^{-3}$ |

$0.0600$ | 0.03000 | $5.70\times {10}^{-3}$ | $6.95\times {10}^{-3}$ |

0.0200 | 0.00040 | $7.00\times {10}^{-5}$ | $9.63\times {10}^{-3}$ |

0.0200 | 0.00010 | $2.00\times {10}^{-5}$ | $9.31\times {10}^{-3}$ |

$\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ |
---|---|---|---|

$0.5$ | $4.7310\times {10}^{-4}$ | $6.8455\times {10}^{-4}$ | $4.00001440$ |

$1.0$ | $4.7310\times {10}^{-4}$ | $6.8455\times {10}^{-4}$ | $4.00058985$ |

$1.5$ | $4.7277\times {10}^{-4}$ | $6.8398\times {10}^{-4}$ | $4.02146973$ |

$2.0$ | $4.5716\times {10}^{-4}$ | $6.5569\times {10}^{-4}$ | $4.58630524$ |

$2.5$ | $9.4000\times {10}^{-7}$ | $8.5000\times {10}^{-7}$ | $7.99999602$ |

$3.0$ | $4.5739\times {10}^{-4}$ | $6.5569\times {10}^{-4}$ | $4.58628884$ |

$3.5$ | $4.7280\times {10}^{-4}$ | $6.8398\times {10}^{-4}$ | $4.02146948$ |

$4.0$ | $4.7310\times {10}^{-4}$ | $6.8454\times {10}^{-4}$ | $4.00058984$ |

$4.5$ | $4.7310\times {10}^{-4}$ | $6.8455\times {10}^{-4}$ | $4.00001440$ |

$5.0$ | $4.7310\times {10}^{-4}$ | $6.8455\times {10}^{-4}$ | $4.00000032$ |

$\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ |
---|---|---|---|

$0.5$ | $1.8921\times {10}^{-4}$ | $2.7382\times {10}^{-4}$ | $4.00001440$ |

$1.0$ | $1.8921\times {10}^{-4}$ | $2.7382\times {10}^{-4}$ | $4.00058985$ |

1.5 | $1.8908\times {10}^{-4}$ | $2.7359\times {10}^{-4}$ | $4.02146966$ |

$2.0$ | $1.8289\times {10}^{-4}$ | $2.6227\times {10}^{-4}$ | $4.58630039$ |

$2.5$ | $1.5000\times {10}^{-7}$ | $1.4000\times {10}^{-4}$ | $7.99999936$ |

$3.0$ | $1.8293\times {10}^{-4}$ | $2.6227\times {10}^{-4}$ | $4.58629383$ |

$3.5$ | $1.8909\times {10}^{-4}$ | $2.7359\times {10}^{-4}$ | $4.02146956$ |

$4.0$ | $1.8921\times {10}^{-4}$ | $2.7382\times {10}^{-4}$ | $4.00058984$ |

$4.5$ | $1.8921\times {10}^{-4}$ | $2.7382\times {10}^{-4}$ | $4.00001440$ |

$5.0$ | $1.8921\times {10}^{-4}$ | $2.7382\times {10}^{-4}$ | $4.00000032$ |

**Table 9.**Comparison of present results for Problem 4.2 with those of Taha et al. [21]. $\left(amplitude=1\right).$

$\mathit{t}$ | $\mathit{h}$ | $\mathbf{\Delta}\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}(\mathbf{Galerkin}\mathbf{Quintic}\mathbf{B}\mathbf{-}\mathbf{spline}\mathbf{Method})$ | ${\mathit{L}}_{\mathit{\infty}}$ [21] |
---|---|---|---|---|

$1.0$ | $0.0500$ | $0.0025$ | $2.36520\times {10}^{-4}$ | $9.60\times {10}^{-4}$ |

$1.0$ | $0.0500$ | $0.0010$ | $9.46000\times {10}^{-5}$ | $1.41\times {10}^{-3}$ |

$1.0$ | $0.6250$ | $0.0071$ | $3.85612\times {10}^{-3}$ | $1.22\times {10}^{-3}$ |

$1.0$ | $0.1300$ | $0.0036$ | $8.44010\times {10}^{-4}$ | $1.41\times {10}^{-3}$ |

$1.6$ | $0.0500$ | $0.0010$ | $9.4460\times {10}^{-5}$ | $1.73\times {10}^{-4}$ |

$1.8$ | $0.0700$ | $0.0700$ | $9.15408\times {10}^{-3}$ | $1.58\times {10}^{-3}$ |

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**MDPI and ACS Style**

Iqbal, A.; Abd Hamid, N.N.; Md. Ismail, A.I.
Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method. *Symmetry* **2019**, *11*, 469.
https://doi.org/10.3390/sym11040469

**AMA Style**

Iqbal A, Abd Hamid NN, Md. Ismail AI.
Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method. *Symmetry*. 2019; 11(4):469.
https://doi.org/10.3390/sym11040469

**Chicago/Turabian Style**

Iqbal, Azhar, Nur Nadiah Abd Hamid, and Ahmad Izani Md. Ismail.
2019. "Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method" *Symmetry* 11, no. 4: 469.
https://doi.org/10.3390/sym11040469